Can you target a revillark with body double when it is reanimated by that reveillark?

Body Double doesn't target anything, you simply choose a creature card. But yes, it can do that. The Reveillark's ability only goes on the Stack after it enters the graveyard, so by the time it resolves and the Body Double enters play, the 'lark has been in the graveyard for a while. (Unless somebody removed it.)

...and if you reanimate a vesuvan shapeshifter and a body double, off a reveillark, can you have the shapeshifter become the body double/body doubles creature?

I don't believe so, given that the Body Double is not yet in-play. However, I would wait for a second opinion on this question.